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On the Topology of the Cambrian Semilattices
For an arbitrary Coxeter group , David Speyer and Nathan Reading defined
Cambrian semilattices as semilattice quotients of the weak order
on induced by certain semilattice homomorphisms. In this article, we define
an edge-labeling using the realization of Cambrian semilattices in terms of
-sortable elements, and show that this is an EL-labeling for every
closed interval of . In addition, we use our labeling to show that
every finite open interval in a Cambrian semilattice is either contractible or
spherical, and we characterize the spherical intervals, generalizing a result
by Nathan Reading.Comment: 20 pages, 5 figure
The M\"obius function of generalized subword order
Let P be a poset and let P* be the set of all finite length words over P.
Generalized subword order is the partial order on P* obtained by letting u \leq
w if and only if there is a subword u' of w having the same length as u such
that each element of u is less than or equal to the corresponding element of u'
in the partial order on P. Classical subword order arises when P is an
antichain, while letting P be a chain gives an order on compositions. For any
finite poset P, we give a simple formula for the Mobius function of P* in terms
of the Mobius function of P. This permits us to rederive in a easy and uniform
manner previous results of Bjorner, Sagan and Vatter, and Tomie. We are also
able to determine the homotopy type of all intervals in P* for any finite P of
rank at most 1.Comment: 29 pages, 4 figures. Incorporates referees' suggestions; to appear in
Advances in Mathematic
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